James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

appendix G The Logarithm Defined as an Integral A51
Notice that the integral that defines ln x is exactly the type of integral discussed in Part 1
of the Fundamental Theorem of Calculus (see Section 5.3). In fact, using that theorem,
we have
d
dx y x 1
1 t dt − 1 x
and so
2
d
dx sln xd − 1 x
We now use this differentiation rule to prove the following properties of the logarithm
function.
3 Laws of Logarithms If x and y are positive numbers and r is a rational number,
then
1. lnsxyd − ln x 1 ln y 2. lnS x yD − ln x 2 ln y 3. lnsx r d − r ln x
Proof
1. Let f sxd − lnsaxd, where a is a positive constant. Then, using Equation 2 and
the Chain Rule, we have
f 9sxd − 1 ax
d
dx saxd − 1 ax ? a − 1 x
Therefore f sxd and ln x have the same derivative and so they must differ by a constant:
lnsaxd − ln x 1 C
Putting x − 1 in this equation, we get ln a − ln 1 1 C − 0 1 C − C. Thus
lnsaxd − ln x 1 ln a
If we now replace the constant a by any number y, we have
2. Using Law 1 with x − 1yy, we have
lnsxyd − ln x 1 ln y
ln 1 y 1 ln y − ln S 1 y ? y D − ln 1 − 0
and so
ln 1 y − 2ln y
Using Law 1 again, we have
lnS x yD − lnSx ? 1 yD − ln x 1 ln 1 y
− ln x 2 ln y
The proof of Law 3 is left as an exercise.
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